Bruce Norris has written a play about race. Not, he says, because he wants to show "why racism in dangerous", but because he likes arguments. And because for all our sensitivity around race today, over who can or can't say what, he believes that the notion of social progress is a myth cleaved to by the left.

"It's like asking, 'What's the progress for dogs?'" splutters Norris, a self-confessed middle-aged, middle-class liberal. "We have certain responsibilities, [but] we don't have progress. I think every generation is one away from a holocaust. We're really lucky terrible racial crimes don't happen more often."

Blunt, spiky, but likable, the Houston-born actor turned playwright is perched in Dominic Cooke's office at the Royal Court, where the new play in question, Clybourne Park, recently opened to five-star reviews. A theatre studies graduate, Norris originally had plans to be a set designer, "but it was too much manual labour". He fixed on writing, his main passion, and bit-part acting ("for the money") and lived in Chicago for 18 years. He has worked closely with the city's lauded Steppenwolf Theatre Company, which produced his last five plays, and moved to New York in 1997.

It was Norris's controversial 2007 hit, The Pain and the Itch, about so-called liberal values, that launched Cooke's own career as artistic director of the Court. Announcing the play, Cooke declared the middle classes "in crisis, confused, and in need of being provoked". Clybourne Park, which Cooke has again directed, is another attempt to do just that.

The play is set in the fictional Chicago neighbourhood originally imagined by Lorraine Hansberry for her 1959 play A Raisin in the Sun, about a black family planning to move into the all-white suburb of Clybourne Park. Norris draws on this, but flips the perspective: in the first act, set in the year of Hansberry's play, we meet Bev and Russ, a middle-aged white couple selling their house to what will be the single black family in the prosperous suburb. With them is Francine, their black maid; her husband Albert; and Hansberry's racist priest, Karl Linder, who begs the couple to reconsider the sale.

"There is a shocking degree of openness in that period to make crass assertions about race," Norris says of his subject matter. "To say, 'Oh, white people are this way but black people are that way.' Today, we have this received etiquette when we're speaking about race, but it is every bit as rigid and ordained as the old vocabulary – we just have a new set of words to talk about similar things." To prove this point, the second act of Norris's play fast-forwards half a century, and turns the tables: a white couple are now trying to move into Clybourne Park, an area long transformed by white flight, and are met by their black neighbours with simmering hostility.

In both acts, the distaste of the locals is terrifically styled, smouldering behind awkward pauses and unfinished sentences. And unlike lots of drama on racism, it's also very funny: the toe-curling exchange of racist jokes in the second act, in particular, has been bringing the house down.

"When it comes to race, everyone is yearning for something to be easier," Norris explains. So anything that addresses the issue without being self-righteous or pious is refreshing? "Exactly." He hesitates. "I mean, I'm sure I am a racist in ways that I'm oblivious to, and it's particularly us privileged whiteys on the left in the US who get themselves in excruciating knots about [prejudice and language]."

Do the middle classes generally have more hang-ups about race? "Well, the middle class is always concerned with looking good, appearing sophisticated. Being tasteful is part of what 'middle class' is all about. It's the middle classes who are always struggling with, 'What's the new word for African-American?' or 'What can I say now?'"

So what about the audience: how do different theatre-goers respond to the play? "The theatre world is weirdly segregated," he says. "In DC [where the play was performed last spring], we got up to 30% black people in the audience, but it was never more than that. This is the US, though," Norris scoffs. "I mean, who goes to theatres? White people."

So far, the audiences in London have been more racially mixed, and rightly so: this is a play for everyone.